Sequentially compact space

In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in ${\displaystyle X}$.

Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.

Examples and properties

The space of all real numbers with the standard topology is not sequentially compact; the sequence ${\displaystyle (s_{n})}$ given by ${\displaystyle s_{n}=n}$ for all natural numbers ${\displaystyle n}$ is a sequence that has no convergent subsequence.

If a space is a metric space, then it is sequentially compact if and only if it is compact.^[1] The first uncountable ordinal with the order topology is an example of a sequentially compact topological space that is not compact. The product of ${\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}}$ copies of the closed unit interval is an example of a compact space that is not sequentially compact.^[2]

Related notions

A topological space ${\displaystyle X}$ is said to be limit point compact if every infinite subset of ${\displaystyle X}$ has a limit point in ${\displaystyle X}$, and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice).

In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.^[3]

There is also a notion of a one-point sequential compactification—the idea is that the non convergent sequences should all converge to the extra point.^[4]

See also

• Bolzano–Weierstrass theorem – Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
• Fréchet–Urysohn space – Property of topological space
• Sequence covering maps
• Sequential space – Topological space characterized by sequences

Notes

1. Willard, 17G, p. 125.
2. Steen and Seebach, Example 105, pp. 125—126.
3. Engelking, General Topology, Theorem 3.10.31
   K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon)
4. Brown, Ronald, "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.

References

• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.